CAIE A-Level Further Math ASematics 1.6.3 Lines and Planes

[Maximum number: 5]

The lines $l_{1}$ and $l_{2}$ have equations

\mathbf{r}=-2 \mathbf{i}-3 \mathbf{j}-5 \mathbf{k}+\lambda(-4 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}) \quad \text { and } \quad \mathbf{r}=2 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k}+\mu(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k})

respectively.

Find the shortest distance between $l_{1}$ and $l_{2}$ .

The plane $\Pi$ contains $l_{1}$ and the point with position vector -i-3 j-4 k.

$\left(\begin{array}{c}2 -2 3\end{array}\right)-\left(\begin{array}{l}-2 -3 -5\end{array}\right)=\left(\begin{array}{l}4 1 8\end{array}\right)$
Finds direction of one line to another.

$\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} -4 3 2 -3 1\end{array}\right|=\left(\begin{array}{c}18 14 6\end{array}\right) \sim\left(\begin{array}{l}9 7 3\end{array}\right)$
Find common perpendicular.

$\frac{1}{\sqrt{139}}\left|\left(\begin{array}{l}4 1 8\end{array}\right) \bullet\left(\begin{array}{l}9 7 3\end{array}\right)\right|=\frac{67}{\sqrt{139}}(=5.68)$
Uses formula for shortest distance.

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CAIE A-Level Further Math AS 1.6.3 Lines and Planes Questions

Exam points

CAIE A-Level Further Math AS 1.6.3 Lines and Planes Questions question 1